Position wave function of two electrons

[Faust90]

Hi,

I want to calculate the position-wave-function of a system of two free electrons with momenta k1 and k2 (vectors).

1. Homework Statement

So, I want to have Psi_(k1,k2)(x1,x2) for a state |k1,k2>

I also know that <k'|k> = (2Pi)^3 Delta(k-k')

The Attempt at a Solution

I tried the following:

Psi(x1,x2)=<x1,x2|k1,k2>=<x1,x2|1|k1,k2>=Integral<x1,x2|k1',k2'><k1',k2'|k1,k2>dk1dk2

The first term <x1,x2|k1',k2'> are the momenta eigenfunction in Space presentation, the second term is the given delta-function:

= Integral Exp(ik1'x1)Exp(ik2'x2) Delta(k1-k1')Delta(k2-k2')
=Exp(ik1x1)Exp(ik2x2)

Is that right?
I'm a bit confused about k1,k2 and k1',k2'Best regards :-)
Faust

 

[DrClaude]

Psi(x1,x2)=<x1,x2|k1,k2>=<x1,x2|1|k1,k2>=Integral<x1,x2|k1',k2'><k1',k2'|k1,k2>dk1dk2

The first term <x1,x2|k1',k2'> are the momenta eigenfunction in Space presentation, the second term is the given delta-function:

= Integral Exp(ik1'x1)Exp(ik2'x2) Delta(k1-k1')Delta(k2-k2')
=Exp(ik1x1)Exp(ik2x2)

Aren't you running in circles here? If you can replace <x1,x2|k1',k2'> by Exp(ik1'x1)Exp(ik2'x2), why can't you do that with <x1,x2|k1,k2>?

If you take definite values of $k$, then the wave function is not localized: the uncertainty on position is infinite. You need to take a more physically acceptable starting point, such as a wave packet centered around $k_1$ and $k_2$. You can then Fourier transform it to get the wave function in position space.

Remember also that the total wave function has to obey the Pauli principle.

 

[Faust90]

Hi,

thanks for your answer :-)

I'm still a bit confused. Let me summary again what I have:

Two electrons characterized by a wave vector k, where I know that the normalization is :<k|k'>=(2Pi)^3 Delta(k-k').
Now I have the state |k1,k2> and I shall construct the space wave function.

My first questions:1. The |k>,|k'> are the eigenvectors of the momentum operator, or?
2. Does the state |k1,k2> mean that I have definite values of k?

 

[DrClaude]

My first questions:

1. The |k>,|k'> are the eigenvectors of the momentum operator, or?
2. Does the state |k1,k2> mean that I have definite values of k?

Yes to both.